Y'know, the square simply doesn't equal any other rectangle.
A square is a very special case. It is, among rectangles, unique. Visually it has dynamic properties that are precisely specific to it. Just WHAT these properties are is subject to some speculation. My opinion and yours may not agree, but different they are nonetheless. That difference provides a very important, yes, the fundamental essential characteristic of the image. One might say that a square is a more passive format, but of course, it also depends upon the graphic content and the way that the visual elements are presented and organized; that could make the passive form internally active, even to the point of violence or chaos. As I mentioned before, designers have told me that they enjoy it because it provides them with lots of options in what they can do with it.
How does the eye move around inside the frame, and how, as designers, do we direct the eye through our graphic choices? A lot of photographers, I think, don't understand that, whether conscious or unconscious of it, they ARE designers, not just "picture takers". That is, even if they are just taking a picture. Design is not a set of skills that most of us are born with; maybe no one is. It is something that must be learned, whether that learning is formal or not. Designing by imitation may work in some individual circumstances, but to do so is simply to commit to a superficial reproduction of others' images, or a class of images. This happens unconsciously, usually. To actively design, it is a whole lot better if one makes decisions wide awake.
Even if they are the same # of sq. units, no two rectangles are the same if their aspect ratio differs. The aspect of the frame is the very basis of what makes the photograph, painting, etc.
If you can't or won't design within a specific bounded space, you can or will do NOTHING in this medium or in any other.